Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 Ã 10<sup>8</sup> and the population of the world as 7 Ã 10<sup>9</sup>, and determine that the world population is more than 20 times larger.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Start Up

20 minutes

When students enter the room, I give them a generic chart (Metric Charts) of metric prefixes and ask them to fill in the blanks. One version of the chart lists standard prefixes for large and small values and the other version separates the large and small value prefixes for the students. I give them the separate charts when I want to minimize the complexity of the problem. I like to break it in pieces, usually starting with the large prefixes, which seem to be easier to visualize. Then they complete the prefixes for smaller values and tape the two charts together. This allows them to build the chart in two pieces and then compare patterns. I like to give the entire chart when I am confident that my students can examine the individual values without being overwhelmed by the long list of values. Again, here is the link to the chart: Metric Charts

The charts are relatively easy to figure out when students see the patterns forming in the cells of a column. I point this out to them if I think they are stuck.

For example,

1 meter = 1

1 dm = .1

1 cm = .01

1 mm = ?

Here they could see that the numbers are getting 10 times smaller with each step and then quickly fill out the values that follow (and vice versa for the larger values).

I give students a few minutes to work on their own before we share our observations. I want students to see the patterns of multiplying or dividing by ten as you travel up and down the chart, but I also want them to see the beautiful symmetry in the numbers and names. For example, Yotta = 10^24 and Yocto = 10^-24. Isn't it wonderful that the furthest reaches of space and the tiniest measurements we can take are equidistant from the center of the scale?

We follow the discussion by adding context to their discoveries. I placed items for each measurement that would help students make sense of this scale. Before the presentation, I give them the most basic reference points.

1 gram is about 1 paperclip (so I let students hold a paperclip to make the physical association)

1 meter is a meter stick (so I show them the stick and place one at each table)

1 liter is 1 kilogram of water (I pass around a liter of water)

The presentation starts with the extremes (a very important technique in math). The idea is to get a sense of the extremes of the scale with the center values of 1 gram, meter and liter in mind. I stress that the specific prefixes are not important. Aside from a few basics (which I list out on the board) like kilo, centi and milli, the students need not worry about the exact order and value of each name. Instead they need to focus on the pattern and structure of the system. Furthermore, they need to understand that the prefixes can be strapped to any value: watts, hertz, etc.

Investigation

20 minutes

I ask students to bring in the walking directions to their favorite place in the city. I ask them to leave out their homes to avoid any socio-economic conflicts. They need to enter the exact address into a map engine and find the directions from our school to their favorite spot. I find it best to explain this and show this. I usually present how to use the Google maps engine at the end of the previous class. They simply need to write out or print out these directions for this activity. As a back up, I pick several locations and print the corresponding walking directions.

"This distance is only about 1.4 miles, but I want to know how many _______ meters this is?"

The ___________ in _________meters represents the prefixes they chose to tackle. As a class I like to take on the prefix of their choice. I ask the to analyze a pair of "symmetric" prefixes. For example, they might choose kilometers and millimeters. Kilometers = 1000 meters and Millimeters = 1/1000 meters.

We discuss how we might go about solving this problem and I ask, "What do you need to know in order to solve this problem?" Here the idea is for students to recognize that we are moving betwen measurement systems and need to know how to convert miles to feet. I give them that information with these two photos from a Google search: Feet and Miles and Feet and Meters

With these numbers we discuss how to find the number of kilometers and millimeters. I ask students to then do the same for their location. They can chose to take on all the prefixes or start by picking a prefix from a bag and then analyze that prefix with its symmetrical partner.

Walk from Penn to Salk

Feet and Meters.png

Feet and Miles.png

Summary

20 minutes

During the summary I ask different students to share their findings. I like to get most groups up to share. So I ask most of them pointed questions like, "how many millimeters is your journey?" I usually ask students who found all the prefixes to go last and preface their presentation with a question. "Bob thought that finding each value was really easy. Can anyone think of why it might be easy to find all the values once you have the meters?"

The heart of this conversation is for students to realize that they simply have to multiply or divide by 10 with each jump up or down the scale. I finish the session by giving all students 5 minutes to write each measurement in scientific notation and help students who are struggling. I often ask students to submit their work if they weren't 100% sure about a value. This gives me a chance to follow up.